Properties

Label 5184.2.d.k.2593.1
Level $5184$
Weight $2$
Character 5184.2593
Analytic conductor $41.394$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5184,2,Mod(2593,5184)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5184, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5184.2593");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(41.3944484078\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2593.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 5184.2593
Dual form 5184.2.d.k.2593.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205i q^{5} -3.46410 q^{7} +O(q^{10})\) \(q-1.73205i q^{5} -3.46410 q^{7} -1.73205i q^{13} +3.00000 q^{17} +2.00000i q^{19} -6.92820 q^{23} +2.00000 q^{25} +8.66025i q^{29} -3.46410 q^{31} +6.00000i q^{35} -8.66025i q^{37} -6.00000 q^{41} +4.00000i q^{43} +3.46410 q^{47} +5.00000 q^{49} +6.92820i q^{53} -12.0000i q^{59} +5.19615i q^{61} -3.00000 q^{65} +10.0000i q^{67} -3.46410 q^{71} +5.00000 q^{73} -6.92820 q^{79} -6.00000i q^{83} -5.19615i q^{85} +15.0000 q^{89} +6.00000i q^{91} +3.46410 q^{95} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{17} + 8 q^{25} - 24 q^{41} + 20 q^{49} - 12 q^{65} + 20 q^{73} + 60 q^{89} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) − 1.73205i − 0.774597i −0.921954 0.387298i \(-0.873408\pi\)
0.921954 0.387298i \(-0.126592\pi\)
\(6\) 0 0
\(7\) −3.46410 −1.30931 −0.654654 0.755929i \(-0.727186\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) − 1.73205i − 0.480384i −0.970725 0.240192i \(-0.922790\pi\)
0.970725 0.240192i \(-0.0772105\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 2.00000i 0.458831i 0.973329 + 0.229416i \(0.0736815\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.92820 −1.44463 −0.722315 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.66025i 1.60817i 0.594515 + 0.804084i \(0.297344\pi\)
−0.594515 + 0.804084i \(0.702656\pi\)
\(30\) 0 0
\(31\) −3.46410 −0.622171 −0.311086 0.950382i \(-0.600693\pi\)
−0.311086 + 0.950382i \(0.600693\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.00000i 1.01419i
\(36\) 0 0
\(37\) − 8.66025i − 1.42374i −0.702313 0.711868i \(-0.747849\pi\)
0.702313 0.711868i \(-0.252151\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.46410 0.505291 0.252646 0.967559i \(-0.418699\pi\)
0.252646 + 0.967559i \(0.418699\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.92820i 0.951662i 0.879537 + 0.475831i \(0.157853\pi\)
−0.879537 + 0.475831i \(0.842147\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 12.0000i − 1.56227i −0.624364 0.781133i \(-0.714642\pi\)
0.624364 0.781133i \(-0.285358\pi\)
\(60\) 0 0
\(61\) 5.19615i 0.665299i 0.943051 + 0.332650i \(0.107943\pi\)
−0.943051 + 0.332650i \(0.892057\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) 10.0000i 1.22169i 0.791748 + 0.610847i \(0.209171\pi\)
−0.791748 + 0.610847i \(0.790829\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.46410 −0.411113 −0.205557 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 0 0
\(73\) 5.00000 0.585206 0.292603 0.956234i \(-0.405479\pi\)
0.292603 + 0.956234i \(0.405479\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.92820 −0.779484 −0.389742 0.920924i \(-0.627436\pi\)
−0.389742 + 0.920924i \(0.627436\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) − 5.19615i − 0.563602i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.0000 1.59000 0.794998 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) 6.00000i 0.628971i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.46410 0.355409
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 13.8564i − 1.37876i −0.724398 0.689382i \(-0.757882\pi\)
0.724398 0.689382i \(-0.242118\pi\)
\(102\) 0 0
\(103\) 10.3923 1.02398 0.511992 0.858990i \(-0.328908\pi\)
0.511992 + 0.858990i \(0.328908\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.0000i 1.74013i 0.492941 + 0.870063i \(0.335922\pi\)
−0.492941 + 0.870063i \(0.664078\pi\)
\(108\) 0 0
\(109\) 8.66025i 0.829502i 0.909935 + 0.414751i \(0.136131\pi\)
−0.909935 + 0.414751i \(0.863869\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 0 0
\(115\) 12.0000i 1.11901i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −10.3923 −0.952661
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 12.1244i − 1.08444i
\(126\) 0 0
\(127\) 13.8564 1.22956 0.614779 0.788700i \(-0.289245\pi\)
0.614779 + 0.788700i \(0.289245\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 6.00000i − 0.524222i −0.965038 0.262111i \(-0.915581\pi\)
0.965038 0.262111i \(-0.0844187\pi\)
\(132\) 0 0
\(133\) − 6.92820i − 0.600751i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −3.00000 −0.256307 −0.128154 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 0 0
\(139\) 2.00000i 0.169638i 0.996396 + 0.0848189i \(0.0270312\pi\)
−0.996396 + 0.0848189i \(0.972969\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 15.0000 1.24568
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 15.5885i 1.27706i 0.769599 + 0.638528i \(0.220456\pi\)
−0.769599 + 0.638528i \(0.779544\pi\)
\(150\) 0 0
\(151\) 20.7846 1.69143 0.845714 0.533637i \(-0.179175\pi\)
0.845714 + 0.533637i \(0.179175\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.00000i 0.481932i
\(156\) 0 0
\(157\) 22.5167i 1.79703i 0.438948 + 0.898513i \(0.355351\pi\)
−0.438948 + 0.898513i \(0.644649\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 24.0000 1.89146
\(162\) 0 0
\(163\) 20.0000i 1.56652i 0.621694 + 0.783260i \(0.286445\pi\)
−0.621694 + 0.783260i \(0.713555\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 17.3205 1.34030 0.670151 0.742225i \(-0.266230\pi\)
0.670151 + 0.742225i \(0.266230\pi\)
\(168\) 0 0
\(169\) 10.0000 0.769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 1.73205i − 0.131685i −0.997830 0.0658427i \(-0.979026\pi\)
0.997830 0.0658427i \(-0.0209736\pi\)
\(174\) 0 0
\(175\) −6.92820 −0.523723
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 18.0000i 1.34538i 0.739923 + 0.672692i \(0.234862\pi\)
−0.739923 + 0.672692i \(0.765138\pi\)
\(180\) 0 0
\(181\) 13.8564i 1.02994i 0.857209 + 0.514969i \(0.172197\pi\)
−0.857209 + 0.514969i \(0.827803\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −15.0000 −1.10282
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3.46410 −0.250654 −0.125327 0.992116i \(-0.539998\pi\)
−0.125327 + 0.992116i \(0.539998\pi\)
\(192\) 0 0
\(193\) 5.00000 0.359908 0.179954 0.983675i \(-0.442405\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.19615i 0.370211i 0.982719 + 0.185105i \(0.0592626\pi\)
−0.982719 + 0.185105i \(0.940737\pi\)
\(198\) 0 0
\(199\) −24.2487 −1.71895 −0.859473 0.511182i \(-0.829208\pi\)
−0.859473 + 0.511182i \(0.829208\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 30.0000i − 2.10559i
\(204\) 0 0
\(205\) 10.3923i 0.725830i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 16.0000i 1.10149i 0.834675 + 0.550743i \(0.185655\pi\)
−0.834675 + 0.550743i \(0.814345\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.92820 0.472500
\(216\) 0 0
\(217\) 12.0000 0.814613
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 5.19615i − 0.349531i
\(222\) 0 0
\(223\) 13.8564 0.927894 0.463947 0.885863i \(-0.346433\pi\)
0.463947 + 0.885863i \(0.346433\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 18.0000i − 1.19470i −0.801980 0.597351i \(-0.796220\pi\)
0.801980 0.597351i \(-0.203780\pi\)
\(228\) 0 0
\(229\) − 8.66025i − 0.572286i −0.958187 0.286143i \(-0.907627\pi\)
0.958187 0.286143i \(-0.0923732\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.0000 0.982683 0.491341 0.870967i \(-0.336507\pi\)
0.491341 + 0.870967i \(0.336507\pi\)
\(234\) 0 0
\(235\) − 6.00000i − 0.391397i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 17.3205 1.12037 0.560185 0.828367i \(-0.310730\pi\)
0.560185 + 0.828367i \(0.310730\pi\)
\(240\) 0 0
\(241\) 5.00000 0.322078 0.161039 0.986948i \(-0.448515\pi\)
0.161039 + 0.986948i \(0.448515\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 8.66025i − 0.553283i
\(246\) 0 0
\(247\) 3.46410 0.220416
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.0000i 1.51487i 0.652913 + 0.757433i \(0.273547\pi\)
−0.652913 + 0.757433i \(0.726453\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −15.0000 −0.935674 −0.467837 0.883815i \(-0.654967\pi\)
−0.467837 + 0.883815i \(0.654967\pi\)
\(258\) 0 0
\(259\) 30.0000i 1.86411i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 10.3923 0.640817 0.320408 0.947279i \(-0.396180\pi\)
0.320408 + 0.947279i \(0.396180\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 15.5885i − 0.950445i −0.879866 0.475223i \(-0.842368\pi\)
0.879866 0.475223i \(-0.157632\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 13.8564i − 0.832551i −0.909239 0.416275i \(-0.863335\pi\)
0.909239 0.416275i \(-0.136665\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −9.00000 −0.536895 −0.268447 0.963294i \(-0.586511\pi\)
−0.268447 + 0.963294i \(0.586511\pi\)
\(282\) 0 0
\(283\) 14.0000i 0.832214i 0.909316 + 0.416107i \(0.136606\pi\)
−0.909316 + 0.416107i \(0.863394\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 20.7846 1.22688
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 19.0526i − 1.11306i −0.830827 0.556531i \(-0.812132\pi\)
0.830827 0.556531i \(-0.187868\pi\)
\(294\) 0 0
\(295\) −20.7846 −1.21013
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.0000i 0.693978i
\(300\) 0 0
\(301\) − 13.8564i − 0.798670i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 9.00000 0.515339
\(306\) 0 0
\(307\) 22.0000i 1.25561i 0.778372 + 0.627803i \(0.216046\pi\)
−0.778372 + 0.627803i \(0.783954\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −34.6410 −1.96431 −0.982156 0.188069i \(-0.939777\pi\)
−0.982156 + 0.188069i \(0.939777\pi\)
\(312\) 0 0
\(313\) 19.0000 1.07394 0.536972 0.843600i \(-0.319568\pi\)
0.536972 + 0.843600i \(0.319568\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.19615i 0.291845i 0.989296 + 0.145922i \(0.0466150\pi\)
−0.989296 + 0.145922i \(0.953385\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.00000i 0.333849i
\(324\) 0 0
\(325\) − 3.46410i − 0.192154i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) − 10.0000i − 0.549650i −0.961494 0.274825i \(-0.911380\pi\)
0.961494 0.274825i \(-0.0886199\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 17.3205 0.946320
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 6.92820 0.374088
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 30.0000i − 1.61048i −0.592946 0.805242i \(-0.702035\pi\)
0.592946 0.805242i \(-0.297965\pi\)
\(348\) 0 0
\(349\) − 34.6410i − 1.85429i −0.374701 0.927146i \(-0.622255\pi\)
0.374701 0.927146i \(-0.377745\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 6.00000i 0.318447i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −34.6410 −1.82828 −0.914141 0.405395i \(-0.867134\pi\)
−0.914141 + 0.405395i \(0.867134\pi\)
\(360\) 0 0
\(361\) 15.0000 0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 8.66025i − 0.453298i
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 24.0000i − 1.24602i
\(372\) 0 0
\(373\) 13.8564i 0.717458i 0.933442 + 0.358729i \(0.116790\pi\)
−0.933442 + 0.358729i \(0.883210\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 15.0000 0.772539
\(378\) 0 0
\(379\) − 2.00000i − 0.102733i −0.998680 0.0513665i \(-0.983642\pi\)
0.998680 0.0513665i \(-0.0163577\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −24.2487 −1.23905 −0.619526 0.784976i \(-0.712675\pi\)
−0.619526 + 0.784976i \(0.712675\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 27.7128i 1.40510i 0.711637 + 0.702548i \(0.247954\pi\)
−0.711637 + 0.702548i \(0.752046\pi\)
\(390\) 0 0
\(391\) −20.7846 −1.05112
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 12.0000i 0.603786i
\(396\) 0 0
\(397\) − 8.66025i − 0.434646i −0.976100 0.217323i \(-0.930268\pi\)
0.976100 0.217323i \(-0.0697324\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9.00000 −0.449439 −0.224719 0.974424i \(-0.572147\pi\)
−0.224719 + 0.974424i \(0.572147\pi\)
\(402\) 0 0
\(403\) 6.00000i 0.298881i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −13.0000 −0.642809 −0.321404 0.946942i \(-0.604155\pi\)
−0.321404 + 0.946942i \(0.604155\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 41.5692i 2.04549i
\(414\) 0 0
\(415\) −10.3923 −0.510138
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 24.0000i 1.17248i 0.810139 + 0.586238i \(0.199392\pi\)
−0.810139 + 0.586238i \(0.800608\pi\)
\(420\) 0 0
\(421\) 22.5167i 1.09739i 0.836021 + 0.548697i \(0.184876\pi\)
−0.836021 + 0.548697i \(0.815124\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) − 18.0000i − 0.871081i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −3.46410 −0.166860 −0.0834300 0.996514i \(-0.526587\pi\)
−0.0834300 + 0.996514i \(0.526587\pi\)
\(432\) 0 0
\(433\) 25.0000 1.20142 0.600712 0.799466i \(-0.294884\pi\)
0.600712 + 0.799466i \(0.294884\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 13.8564i − 0.662842i
\(438\) 0 0
\(439\) −10.3923 −0.495998 −0.247999 0.968760i \(-0.579773\pi\)
−0.247999 + 0.968760i \(0.579773\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 30.0000i 1.42534i 0.701498 + 0.712672i \(0.252515\pi\)
−0.701498 + 0.712672i \(0.747485\pi\)
\(444\) 0 0
\(445\) − 25.9808i − 1.23161i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 10.3923 0.487199
\(456\) 0 0
\(457\) −13.0000 −0.608114 −0.304057 0.952654i \(-0.598341\pi\)
−0.304057 + 0.952654i \(0.598341\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −6.92820 −0.321981 −0.160990 0.986956i \(-0.551469\pi\)
−0.160990 + 0.986956i \(0.551469\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) − 34.6410i − 1.59957i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 4.00000i 0.183533i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 10.3923 0.474837 0.237418 0.971408i \(-0.423699\pi\)
0.237418 + 0.971408i \(0.423699\pi\)
\(480\) 0 0
\(481\) −15.0000 −0.683941
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 17.3205i − 0.786484i
\(486\) 0 0
\(487\) 3.46410 0.156973 0.0784867 0.996915i \(-0.474991\pi\)
0.0784867 + 0.996915i \(0.474991\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 24.0000i − 1.08310i −0.840667 0.541552i \(-0.817837\pi\)
0.840667 0.541552i \(-0.182163\pi\)
\(492\) 0 0
\(493\) 25.9808i 1.17011i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) 28.0000i 1.25345i 0.779240 + 0.626726i \(0.215605\pi\)
−0.779240 + 0.626726i \(0.784395\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 34.6410 1.54457 0.772283 0.635278i \(-0.219115\pi\)
0.772283 + 0.635278i \(0.219115\pi\)
\(504\) 0 0
\(505\) −24.0000 −1.06799
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 27.7128i 1.22835i 0.789170 + 0.614174i \(0.210511\pi\)
−0.789170 + 0.614174i \(0.789489\pi\)
\(510\) 0 0
\(511\) −17.3205 −0.766214
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 18.0000i − 0.793175i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 4.00000i 0.174908i 0.996169 + 0.0874539i \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10.3923 −0.452696
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 10.3923i 0.450141i
\(534\) 0 0
\(535\) 31.1769 1.34790
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) − 8.66025i − 0.372333i −0.982518 0.186167i \(-0.940394\pi\)
0.982518 0.186167i \(-0.0596065\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 15.0000 0.642529
\(546\) 0 0
\(547\) 2.00000i 0.0855138i 0.999086 + 0.0427569i \(0.0136141\pi\)
−0.999086 + 0.0427569i \(0.986386\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −17.3205 −0.737878
\(552\) 0 0
\(553\) 24.0000 1.02058
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5.19615i 0.220168i 0.993922 + 0.110084i \(0.0351120\pi\)
−0.993922 + 0.110084i \(0.964888\pi\)
\(558\) 0 0
\(559\) 6.92820 0.293032
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 30.0000i 1.26435i 0.774826 + 0.632175i \(0.217837\pi\)
−0.774826 + 0.632175i \(0.782163\pi\)
\(564\) 0 0
\(565\) 25.9808i 1.09302i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3.00000 −0.125767 −0.0628833 0.998021i \(-0.520030\pi\)
−0.0628833 + 0.998021i \(0.520030\pi\)
\(570\) 0 0
\(571\) − 40.0000i − 1.67395i −0.547243 0.836974i \(-0.684323\pi\)
0.547243 0.836974i \(-0.315677\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −13.8564 −0.577852
\(576\) 0 0
\(577\) −5.00000 −0.208153 −0.104076 0.994569i \(-0.533189\pi\)
−0.104076 + 0.994569i \(0.533189\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 20.7846i 0.862291i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.0000i 0.495293i 0.968850 + 0.247647i \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) 0 0
\(589\) − 6.92820i − 0.285472i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −39.0000 −1.60154 −0.800769 0.598973i \(-0.795576\pi\)
−0.800769 + 0.598973i \(0.795576\pi\)
\(594\) 0 0
\(595\) 18.0000i 0.737928i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 24.2487 0.990775 0.495388 0.868672i \(-0.335026\pi\)
0.495388 + 0.868672i \(0.335026\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 19.0526i − 0.774597i
\(606\) 0 0
\(607\) 17.3205 0.703018 0.351509 0.936185i \(-0.385669\pi\)
0.351509 + 0.936185i \(0.385669\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 6.00000i − 0.242734i
\(612\) 0 0
\(613\) 41.5692i 1.67896i 0.543387 + 0.839482i \(0.317142\pi\)
−0.543387 + 0.839482i \(0.682858\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.00000 0.120775 0.0603877 0.998175i \(-0.480766\pi\)
0.0603877 + 0.998175i \(0.480766\pi\)
\(618\) 0 0
\(619\) − 28.0000i − 1.12542i −0.826656 0.562708i \(-0.809760\pi\)
0.826656 0.562708i \(-0.190240\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −51.9615 −2.08179
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 25.9808i − 1.03592i
\(630\) 0 0
\(631\) 34.6410 1.37904 0.689519 0.724268i \(-0.257822\pi\)
0.689519 + 0.724268i \(0.257822\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 24.0000i − 0.952411i
\(636\) 0 0
\(637\) − 8.66025i − 0.343132i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −33.0000 −1.30342 −0.651711 0.758468i \(-0.725948\pi\)
−0.651711 + 0.758468i \(0.725948\pi\)
\(642\) 0 0
\(643\) 26.0000i 1.02534i 0.858586 + 0.512670i \(0.171344\pi\)
−0.858586 + 0.512670i \(0.828656\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −24.2487 −0.953315 −0.476658 0.879089i \(-0.658152\pi\)
−0.476658 + 0.879089i \(0.658152\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) −10.3923 −0.406061
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 18.0000i − 0.701180i −0.936529 0.350590i \(-0.885981\pi\)
0.936529 0.350590i \(-0.114019\pi\)
\(660\) 0 0
\(661\) 46.7654i 1.81896i 0.415745 + 0.909481i \(0.363521\pi\)
−0.415745 + 0.909481i \(0.636479\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −12.0000 −0.465340
\(666\) 0 0
\(667\) − 60.0000i − 2.32321i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −35.0000 −1.34915 −0.674575 0.738206i \(-0.735673\pi\)
−0.674575 + 0.738206i \(0.735673\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 13.8564i 0.532545i 0.963898 + 0.266272i \(0.0857921\pi\)
−0.963898 + 0.266272i \(0.914208\pi\)
\(678\) 0 0
\(679\) −34.6410 −1.32940
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 30.0000i − 1.14792i −0.818884 0.573959i \(-0.805407\pi\)
0.818884 0.573959i \(-0.194593\pi\)
\(684\) 0 0
\(685\) 5.19615i 0.198535i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 26.0000i 0.989087i 0.869153 + 0.494543i \(0.164665\pi\)
−0.869153 + 0.494543i \(0.835335\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.46410 0.131401
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 25.9808i 0.981280i 0.871362 + 0.490640i \(0.163237\pi\)
−0.871362 + 0.490640i \(0.836763\pi\)
\(702\) 0 0
\(703\) 17.3205 0.653255
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 48.0000i 1.80523i
\(708\) 0 0
\(709\) − 1.73205i − 0.0650485i −0.999471 0.0325243i \(-0.989645\pi\)
0.999471 0.0325243i \(-0.0103546\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −41.5692 −1.55027 −0.775135 0.631795i \(-0.782318\pi\)
−0.775135 + 0.631795i \(0.782318\pi\)
\(720\) 0 0
\(721\) −36.0000 −1.34071
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 17.3205i 0.643268i
\(726\) 0 0
\(727\) 31.1769 1.15629 0.578144 0.815935i \(-0.303777\pi\)
0.578144 + 0.815935i \(0.303777\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 12.0000i 0.443836i
\(732\) 0 0
\(733\) 34.6410i 1.27950i 0.768585 + 0.639748i \(0.220961\pi\)
−0.768585 + 0.639748i \(0.779039\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 10.0000i 0.367856i 0.982940 + 0.183928i \(0.0588813\pi\)
−0.982940 + 0.183928i \(0.941119\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 17.3205 0.635428 0.317714 0.948187i \(-0.397085\pi\)
0.317714 + 0.948187i \(0.397085\pi\)
\(744\) 0 0
\(745\) 27.0000 0.989203
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 62.3538i − 2.27836i
\(750\) 0 0
\(751\) 17.3205 0.632034 0.316017 0.948753i \(-0.397654\pi\)
0.316017 + 0.948753i \(0.397654\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 36.0000i − 1.31017i
\(756\) 0 0
\(757\) 20.7846i 0.755429i 0.925922 + 0.377715i \(0.123290\pi\)
−0.925922 + 0.377715i \(0.876710\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 21.0000 0.761249 0.380625 0.924730i \(-0.375709\pi\)
0.380625 + 0.924730i \(0.375709\pi\)
\(762\) 0 0
\(763\) − 30.0000i − 1.08607i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −20.7846 −0.750489
\(768\) 0 0
\(769\) 5.00000 0.180305 0.0901523 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 8.66025i − 0.311488i −0.987797 0.155744i \(-0.950223\pi\)
0.987797 0.155744i \(-0.0497774\pi\)
\(774\) 0 0
\(775\) −6.92820 −0.248868
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 12.0000i − 0.429945i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 39.0000 1.39197
\(786\) 0 0
\(787\) − 28.0000i − 0.998092i −0.866575 0.499046i \(-0.833684\pi\)
0.866575 0.499046i \(-0.166316\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 51.9615 1.84754
\(792\) 0 0
\(793\) 9.00000 0.319599
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 19.0526i 0.674876i 0.941348 + 0.337438i \(0.109560\pi\)
−0.941348 + 0.337438i \(0.890440\pi\)
\(798\) 0 0
\(799\) 10.3923 0.367653
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) − 41.5692i − 1.46512i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −27.0000 −0.949269 −0.474635 0.880183i \(-0.657420\pi\)
−0.474635 + 0.880183i \(0.657420\pi\)
\(810\) 0 0
\(811\) − 20.0000i − 0.702295i −0.936320 0.351147i \(-0.885792\pi\)
0.936320 0.351147i \(-0.114208\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 34.6410 1.21342
\(816\) 0 0
\(817\) −8.00000 −0.279885
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8.66025i 0.302245i 0.988515 + 0.151122i \(0.0482888\pi\)
−0.988515 + 0.151122i \(0.951711\pi\)
\(822\) 0 0
\(823\) 17.3205 0.603755 0.301877 0.953347i \(-0.402387\pi\)
0.301877 + 0.953347i \(0.402387\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 12.0000i − 0.417281i −0.977992 0.208640i \(-0.933096\pi\)
0.977992 0.208640i \(-0.0669038\pi\)
\(828\) 0 0
\(829\) − 34.6410i − 1.20313i −0.798823 0.601566i \(-0.794544\pi\)
0.798823 0.601566i \(-0.205456\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 15.0000 0.519719
\(834\) 0 0
\(835\) − 30.0000i − 1.03819i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 34.6410 1.19594 0.597970 0.801518i \(-0.295974\pi\)
0.597970 + 0.801518i \(0.295974\pi\)
\(840\) 0 0
\(841\) −46.0000 −1.58621
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 17.3205i − 0.595844i
\(846\) 0 0
\(847\) −38.1051 −1.30931
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 60.0000i 2.05677i
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −15.0000 −0.512390 −0.256195 0.966625i \(-0.582469\pi\)
−0.256195 + 0.966625i \(0.582469\pi\)
\(858\) 0 0
\(859\) − 50.0000i − 1.70598i −0.521929 0.852989i \(-0.674787\pi\)
0.521929 0.852989i \(-0.325213\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 27.7128 0.943355 0.471678 0.881771i \(-0.343649\pi\)
0.471678 + 0.881771i \(0.343649\pi\)
\(864\) 0 0
\(865\) −3.00000 −0.102003
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 17.3205 0.586883
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 42.0000i 1.41986i
\(876\) 0 0
\(877\) − 29.4449i − 0.994282i −0.867670 0.497141i \(-0.834383\pi\)
0.867670 0.497141i \(-0.165617\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) − 10.0000i − 0.336527i −0.985742 0.168263i \(-0.946184\pi\)
0.985742 0.168263i \(-0.0538159\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −20.7846 −0.697879 −0.348939 0.937145i \(-0.613458\pi\)
−0.348939 + 0.937145i \(0.613458\pi\)
\(888\) 0 0
\(889\) −48.0000 −1.60987
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.92820i 0.231843i
\(894\) 0 0
\(895\) 31.1769 1.04213
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 30.0000i − 1.00056i
\(900\) 0 0
\(901\) 20.7846i 0.692436i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 24.0000 0.797787
\(906\) 0 0
\(907\) − 8.00000i − 0.265636i −0.991140 0.132818i \(-0.957597\pi\)
0.991140 0.132818i \(-0.0424025\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 34.6410 1.14771 0.573854 0.818958i \(-0.305448\pi\)
0.573854 + 0.818958i \(0.305448\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 20.7846i 0.686368i
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 6.00000i 0.197492i
\(924\) 0 0
\(925\) − 17.3205i − 0.569495i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −21.0000 −0.688988 −0.344494 0.938789i \(-0.611949\pi\)
−0.344494 + 0.938789i \(0.611949\pi\)
\(930\) 0 0
\(931\) 10.0000i 0.327737i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −35.0000 −1.14340 −0.571700 0.820463i \(-0.693716\pi\)
−0.571700 + 0.820463i \(0.693716\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 43.3013i − 1.41158i −0.708421 0.705791i \(-0.750592\pi\)
0.708421 0.705791i \(-0.249408\pi\)
\(942\) 0 0
\(943\) 41.5692 1.35368
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 18.0000i − 0.584921i −0.956278 0.292461i \(-0.905526\pi\)
0.956278 0.292461i \(-0.0944741\pi\)
\(948\) 0 0
\(949\) − 8.66025i − 0.281124i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −39.0000 −1.26333 −0.631667 0.775240i \(-0.717629\pi\)
−0.631667 + 0.775240i \(0.717629\pi\)
\(954\) 0 0
\(955\) 6.00000i 0.194155i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 10.3923 0.335585
\(960\) 0 0
\(961\) −19.0000 −0.612903
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 8.66025i − 0.278783i
\(966\) 0 0
\(967\) −51.9615 −1.67097 −0.835485 0.549513i \(-0.814813\pi\)
−0.835485 + 0.549513i \(0.814813\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 6.00000i − 0.192549i −0.995355 0.0962746i \(-0.969307\pi\)
0.995355 0.0962746i \(-0.0306927\pi\)
\(972\) 0 0
\(973\) − 6.92820i − 0.222108i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 20.7846 0.662926 0.331463 0.943468i \(-0.392458\pi\)
0.331463 + 0.943468i \(0.392458\pi\)
\(984\) 0 0
\(985\) 9.00000 0.286764
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 27.7128i − 0.881216i
\(990\) 0 0
\(991\) −51.9615 −1.65061 −0.825306 0.564686i \(-0.808997\pi\)
−0.825306 + 0.564686i \(0.808997\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 42.0000i 1.33149i
\(996\) 0 0
\(997\) − 5.19615i − 0.164564i −0.996609 0.0822819i \(-0.973779\pi\)
0.996609 0.0822819i \(-0.0262208\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5184.2.d.k.2593.1 yes 4
3.2 odd 2 5184.2.d.d.2593.3 yes 4
4.3 odd 2 inner 5184.2.d.k.2593.2 yes 4
8.3 odd 2 inner 5184.2.d.k.2593.4 yes 4
8.5 even 2 inner 5184.2.d.k.2593.3 yes 4
12.11 even 2 5184.2.d.d.2593.4 yes 4
24.5 odd 2 5184.2.d.d.2593.1 4
24.11 even 2 5184.2.d.d.2593.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5184.2.d.d.2593.1 4 24.5 odd 2
5184.2.d.d.2593.2 yes 4 24.11 even 2
5184.2.d.d.2593.3 yes 4 3.2 odd 2
5184.2.d.d.2593.4 yes 4 12.11 even 2
5184.2.d.k.2593.1 yes 4 1.1 even 1 trivial
5184.2.d.k.2593.2 yes 4 4.3 odd 2 inner
5184.2.d.k.2593.3 yes 4 8.5 even 2 inner
5184.2.d.k.2593.4 yes 4 8.3 odd 2 inner